An important unit consists of two components placed in parallel. Only one component is operated at a time, but both components are kept operational as often as possible by repairing them as needed. An operating component breaks down in a given period with probability $0\mathrm{.}3.$ When this occurs, the parallel component takes over, if it is operational, at the beginning of the next period. Due to the limitations of the repair personnel's capabilities, only one component can be repaired at a time.$($a$)$ Formulate the condition of this unit as a Markov Chain by defining its states space and giving its one$-$step transition matrix.$($b$)$ What is the probability that the unit will be operable after $2$ periods?$($c$)$ If it costs $$50000$ per period when the unit is inoperable and zero otherwise, what is the expected average cost per period?